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The unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\}$$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w$$ setting it to zero, and solving for $w$. This gives $$w_{mv,unc}\equiv\frac{1}{\lambda}\... 11 After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected returns (Black-Litterman) as well as with the in-house views of Chief Investment Officer Burton Malkiel’s team. In addition, Wealthfront sets minimum and ... 11 Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero. Discussion A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ... 10 The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow. There isn't necessarily a difference between these quantities in a long-only ... 8 Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ... 8 To clarify notation, you have an universe of n=2000 \space stocks and two portfolio vectors \mathbf{a},\mathbf{b}\in\mathbb{R}^{n} with \left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1. Further, you have Estimators for the true Variance \operatorname{Var}\left[\mathbf{a}\right] resp. \operatorname{Var}\left[\mathbf{b}\right] and the ... 8 Portfolio optimalisation depends heavily on the estimation of the moments (and therefore has HUGE estimation uncertainty). Even though it's useful for comparing and analysing different existing strategies, I think practitioners are moving more towards the usage of factor portfolios for the strategies themselves (e.g. Fama-French). Also because the ... 8 The best explanation I have seen so far is the so-called Adaptive Market Hypothesis by Andrew Lo: The adaptive market hypothesis, as proposed by Andrew Lo, is an attempt to reconcile economic theories based on the efficient market hypothesis (which implies that markets are efficient) with behavioral economics, by applying the principles of ... 8 CAPM states that the expected return of any given asset should equal ER_i=R_f+β_i (R_m-R_f), with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that E[(R_m-R_f )]>0). Every individual stock has some idiosyncratic risk in addition to ... 7 The term in sample and out of sample are commonly used in any kind of optimization or fitting methods (MVO is just a particular case). When you make the optimization, you compute optimal parameters (usually the weights of the optimal portfolio in asset allocation) over a given data sample, for example, the returns of the securities of the portfolio for the ... 7 Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf 7 This is indeed a subtle point. What is generally meant with this statement is that correlation is going up in bear markets, so it is not so much the "turmoil" part (i.e. volatility per se) but the "trend" (i.e. negative in this case) part. Putting it another way is that when you control for volatility not the correlation but the covariance (which is the part ... 7 As a practitioner, I have worked on the following Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There are other constraints such as No short selling Max amount you can buy is X% of Max outstanding amount in market Maximum exposure to a perticular country , ... 7 if you take the variance of a single asset it scales as a quadratic,$$ var(\lambda X) = \lambda^2 var(X) $$so it's not surprising that the general case gives a quadratic form. 7 A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few more recent articles about portfolio theory, in no particular order (all accessible online): Jorion: Bayes-Stein Estimation for Portfolio Analysis, JFQA, 1986 ... 6 Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to "0" if the ... 6 Seems like a small mistake in the last equation. It should read \Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\right], which is not equivalent to your result. 6 concerning your first question: the derivative does not disappear: \sigma(R_p) contains the square root. To be more precise, set$$ \sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)^2 + 2w_1w_2\text{Cov}(R_1, R_2)}. Then we get using the chain rule: \begin{align} \frac{\partial\sigma(R_p)}{\partial w_1} &= \frac 12 \cdot \biggl(\... 6 1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. So for eg stocks at 20 vol, that's a +/-40% 1y 95% confidence interval around your sample mean ;-) With 100 years of data, that's still +/-4%! Which is in-line ... 6 In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), others (including Markowitz himself) thought this was not realistic (he thought if you have 1 dollar you cannot both buy 1 dollar worth of stock and also short 1 ... 5 Typical risk aversion levels lie between one and ten. See pages 11f. in the following paper: Preferences by Andrew Ang EDIT: Unfortunately the paper doesn't seem to be available online anymore. The final source is the following book: Asset Management: A Systematic Approach to Factor Investing (Financial Management Association Survey and Synthesis) 1st ... 5 The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement. In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric ... 5 There is a great deal of misinformation and out-of-date information on this site. Many of the references in this discussion and elsewhere have serious research flaws. The Michaud efficient frontier was invented and patented by Robert Michaud and Richard Michaud, U.S. patent # 6,003,018. The alternatives discussed here are not patented nor in many cases ... 5 Any explanations? Yes. Within each asset category we find that stocks may be: Unattractively underperforming the category norm Attractive as they meet the expected norm Unsustainable as their returns exceed the category norm and may suffer mean reversion By focusing on low variance, we exclude type (3) stocks that damage portfolio performance through high ... 5 Accurately stated: Diversification helps during turmoil, but helps less as what would be expected by using w^T \Omega w as the portfolio variance where the off-diagonal covariances are estimated during tranquil periods. This is because correlations and covariances change during turmoil, typically increasing. This reduces the benefit of diversification ... 5 This optimization is trivial w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases}  That is to say, when you optimize only one weight will be nonzero. That's because these ratios incorporate no notion of distributional width, and therefore do not reward diversification. With no concentration ...

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The weak EMH states that it is impossible to earn an excess return given publicly known information such as past prices. Clearly, different securities have different expected returns. For example: the bond and the stock of one company or a security that generates twice the return of another one. This difference in expected return is explained by a ...

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There are plenty of books on portfolio issues built according to formula "some theory + some R code (or Matlab, or S - which is very similar to R)". See for example Pfaff B. Financial Risk Modelling and Portfolio Optimization with R.// 2013. Best M.J. Portfolio Optimization. Chapman & Hall, 2010. Würtz D. et al. Portfolio Optimization with R/Rmetrics. ...

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Sharpe's 1966 equation had $R_b$ defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent. If we refer to the original definitions, then that is the primary difference - Sharpe's ratio looks at reward/risk of the excess return for an asset over the risk-free rate while the ...

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Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it is a chance to discuss today's approaches. A nice approach that is very up-to-date where mementum investing seems very fashionable is the following: Momentum ...

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